Yau-Tian-Donaldson conjecture

The Yau-Tian-Donaldson conjecture relates to the existence of special geometric structures, called Kähler–Einstein metrics, on complex geometric spaces known as Fano manifolds. It predicts that such a metric exists if and only if the manifold is stable in a specific algebraic sense, called K-stability. In essence, the conjecture connects a geometric condition (existence of a canonical shape) with an algebraic stability property, providing a criterion to determine when these special metrics can be found. This deep link unifies geometry and algebra, guiding mathematicians in understanding the shape and structure of complex spaces.